Discrete Mathematics
Ch1.1
Propositional Logic
Dr. Talal Bonny
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Definition of a Proposition
A proposition (p, q, r, …) is simply a declarative statement
with a definite meaning, having a truth value that’s
either true (T) or false (F) (never both, neither, or
somewhere in between).
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Examples on Propositions
“It is raining.” (Given a situation.)
“UAE exports oil”
“1 + 2 = 3”
The following are NOT propositions:
“Who’s there?” (interrogative, question)
“Just do it!” (command)
“1 + 2” (expression with a non-true/false value)
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Propositional variables
We use propositional variables to refer to propositions
o Usually are lower case letters starting with p (i.e. p, q, r, s,
etc.)
o A propositional variable can have one of two values: true (T)
or false (F)
A proposition can be…
o A single variable: p
o An operation of multiple variables: p∧(q∨¬r)
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Propositional Logic
Propositional Logic is the logic of compound statements
built from simpler statements using Boolean
connectives.
Applications:
Design of digital electronic circuits.
Expressing conditions in programs.
Queries to databases & search engines.
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Operators / Connectives
An operator or connective combines one or more
operand expressions into a larger expression.
It is useful to establish analogy with traditional math
(numbers):
o E.g., “+ - * /” in numeric expression
 Unary operators take one operand (e.g., -3);
 Binary operators take two operands (eg 3 × 4).
o Propositional or Boolean operators operate on
propositions or truth values instead of on numbers.
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Logical Operators
∧ Logical And
P ∧ Q, (P ∧ Q) ∧ R, …
∨ Logical Or
P ∨ Q, P ∨ (Q ∨ R), …
¬ Logical Not
¬P, ((¬P) ∧ Q), …
→ Logical Implication P→Q, (P∧Q)→Q, …
↔ Logical Equivalence P↔Q, (P∧Q)↔(Q∧P)
⊗ Exclusive Or
P ⊗ Q, (P ∧ Q) ⊗ R, …
Logical operators provide ways to combine
propositions for calculation
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Introduction to Logical Operators
In the following examples,
o p : “Today is Friday”
o q : “Today is the beginning of Ramadan”
o r:x>0
o s : x < 10
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Logical conjunction operators: And
An “and” operation is true if both operands are true
Symbol: ∧
o It’s like the ‘A’ in And
p ∧ q =
r ∧ s =
Note that a conjunction
p 1 ∧ p2 ∧ … ∧ pn
of n propositions will have
p
q
p∧q
F
F
F
T
F
F
T
F
F
T
T
T
rows in its truth table.
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Logical disjunction operators: Or
An “or” operation is true if either operands are true
Symbol: ∨
P ∨ q =
So this operation is
p
q
p∨q
F
F
F
T
F
T
T
F
T
T
T
T
also called “inclusive or”, because it includes the
possibility that both p and q are true.
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Logical operators: Not
A “not” operation is true if operand is false. It switches
(negates) the truth value
Symbol: ¬ or ~
p
¬p
¬p =
F
T
F
¬r =
T
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A Simple Exercise
Let p = “It rained last night”,
q = “The sprinklers came on last night,”
r = “The grass was wet this morning.”
Translate each of the following into English:
¬p
=
r ∧ ¬p
=
“Either the grass wasn’t wet this
morning, or it rained last night, or =
the sprinklers came on last night.”
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Natural Language is Unclear
Note that in English “or” is by itself corresponds to one of
the two cases or “both”.
If the “both” is excluded, then exclusive or ⊕ is used
“Pat is a singer or
Pat is a writer.” “Pat is a man or
Pat is a woman.” -
p
F
F
T
T
q p or q
F
F
T
T
F
T
T undef.
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Logical operators: Exclusive Or
An exclusive or operation is true if one of the operands
are true, but false if both are true
Symbol: ⊕
p
q
p⊕q
Often called XOR
F
F
To write it using logical operators
F
T
F
T
and based on ∨ table
T
F
p⊕q ≡ (p ∨ q) ∧ ¬(p ∧ q)
T
T
T
F
o p = “Today is Friday”
o q = “Today is the beginning of Ramadan”
p⊕q =
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Examples
Assume P = x > 0, Q = y > 0
If P ⊗ Q is True, in which quadrant should be (x, y)?
If P ∧ Q is True, in which quadrant should be (x, y)?
If P ∨ Q is True, in which quadrant should be (x, y)?
If ¬ P is True, in which quadrant should be (x, y)?
y
Quadrant 2 Quadrant 1
x < 0, y > 0 x > 0, y > 0
x
Quadrant 3 Quadrant 4
x < 0, y < 0 x > 0, y < 0
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Inclusive Or versus Exclusive Or
Do these sentences mean inclusive or exclusive or?
o Experience with C++ or Java is required
o Lunch includes soup or salad
o To enter the country, you need a passport or a driver’s license
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Conditional Statement
 Let p and q be propositions. The conditional statement p→q is
false when p is true and q is false, and true otherwise
 A conditional means “if p then q”
p→q
p
q
 Symbol: →
T
F
F
o p is called the hypothesis
o q is called the conclusion
o p→q is called implication
F
T
T
F
T
F
T
T
T
Note that if p is false, then
the conditional statement is
true regardless of whether q
is true or false
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Conditional Statement Example 1
 p = “the student is studying engineering”
 q = “s/he is smart”
 p→q =
o “If the student is studying engineering,
then s/he is smart” TRUE
o “If the student is not studying
engineering, then s/he is smart” TRUE
 He is studying business for ex.
o “If the student is not studying
engineering, then s/he is not smart” TRUE
p
q
p→q
F
F
F
T
T
F
T
T
F
T
T
T
 He is not studying at all
o “If the student is studying engineering, then
s/he is not smart” FALSE.
 Not correct because he can not continue studying
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Conditional Statement Example 2
 p = “you get 100% on the final”
 q = “you will get an A”
 p→q =
o “If you get 100% on the final,
then you will get an A” TRUE
o “If you did not get 100% on the final,
then you may get an A” TRUE
 (if you get 90% and above)
o “If you did not get 100% on the final,
then you may not get an A” TRUE
p
q
p→q
F
F
F
T
T
F
T
T
F
T
T
T
 (if you get less than 90%)
o “If you get 100% on the final,
then you will not get an A” FALSE
not allowed, you will feel cheated
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Conditional Statement
 To write it using logical operator:
p→q=¬p∨ q
Ex.1: The student is not studying engineering or s/he is smart
Ex.2: you did not get 100% on the final or you will get an A
Alternate ways of stating a conditional statements:
o
o
o
o
o
o
o
p implies q
If p, q
p only if q
p is sufficient for q
q if p
q whenever p
q is necessary for p
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Conditional Statement (Cont.)
p is not sufficient for q = ¬ (p→q)
p q
¬p
¬q
Conditional
Inverse
p→q
¬p→¬q
Converse Contrapositive
q→p
¬q→¬p
F F
F T
T F
T T
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Bi-conditional Statement
P↔q: a bi-conditional means “p if and only if q”
Symbol: ↔
Alternatively, it means
“(if p then q) and (if q then p)”
Let p = “the student is studying engineering”
and q = “s/he is smart”, then p↔q means
o “the student is studying engineering if and only if
s/he is smart”
o Alternatively, it means “If the student is smart, then
s/he is studying engineering” and if the student is studying
engineering then s/he is smart
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Bi-conditional Statement
 To write it using logical operator:
p→q ∧ q→p = (¬p∨ q) ∧ (¬q∨ p)
Alternate ways of stating a conditional statements:
o “p is necessary and sufficient for q”
o “if p then q, and conversely”
o “p iff q”
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Boolean operators summary
p
q
F
F
F
T
T
F
T
T
not
not
and
or
xor
conditional
Bi-conditional
¬p
¬q
p∧q
p∨q
p⊕q
p→q
p↔q
Learn what they mean, don’t just memorize the table!
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Precedence of operators
Just as in algebra, operators have precedence
o 4+3*2 =
o 4+(3*2) = 10 but not (4+3)*2 = 14
Precedence order
(from highest to lowest):
Example:
p ∨ q ∧ ¬r → s ↔ t
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Bit Operations
 The word bit comes from binary digit
 Boolean values can be represented as 1 (true) and 0 (false)
o By replacing true by a one
o and false by a zero in the truth table
 Truth tables for AND, OR, XOR
 Boolean variable in programming language or database
systems can be stored in 1 bit
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Bit Operations
 A bit string is a series of Boolean values. The length of this string
is the number of bits
o in the string 10110100 is eight Boolean values in one string
 We can then do operations on these Boolean strings
o Each column is its own Boolean operation
 Find the bitwise OR, bitwise AND, and bitwise XOR of the bit
strings 01011010 and 10110100
01011010
∧10110100
01011010
∨10110100
01011010
⊕10110100
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Examples
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Apply the alternate ways of the conditional statement on
the previous example:
p = “you get 100% on the final”
q = “you will get an A”
Alternate ways of stating a conditional statements:
o If p, q
o p only if q
o q if p
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Example1
 Question 7 from Rosen, p. 17 (Translating English Sentences )
o p = “It is below freezing”
o q = “It is snowing”
Write these propositions using p and q and logical connectives
 It is below freezing and it is snowing
 It is below freezing but not snowing
 It is not below freezing and it is not snowing
 It is either snowing or below freezing (or both)
 If it is below freezing, it is also snowing
 That it is below freezing is necessary and
sufficient for it to be snowing
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Example 2
Express the following as a logical Expression:“I have
neither given nor received help on this exam”
Let p = “I have given help on this exam”
Let q = “I have received help on this exam”
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Example 3
Express the following as a logical Expression: You can
access the Internet from campus only if you are a
computer engineer major or you are not a freshman.
Let a represents "You can access the Internet from
campus"
Let c represents "You are a computer engineer
major," and
Let f represents "You are a freshman,“
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Example 4
Construct the truth table of the logical Expression:
((P ∨ Q)∧((¬P)→Q))
P
Q
False
False
False
True
True
False
True
True
(P ∨ Q)
(¬P)
((¬P)→Q)
((P∨Q)∧((¬P)→Q))
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Example 5
Construct the truth table of the logical Expression:
(((¬P)∧Q)∧((¬P)→Q))
P
Q
(¬P)
((¬P)∧Q) ((¬P)→Q)
( (((¬P)∧Q) ∧((¬P)→Q))
False
False
False
False
True
True
True
False
False
True
True
False
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Example 7: Bit Operations
 Evaluate the following expressions
o (11011 ∨ 01010) ∧ (10001 ∨ 11011)
o 11011 ∨ 01010 ∧ 10001 ∨ 11011
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